• 


LIBRARY 

OF  THE 

University  of  California. 

GIFT  OF 

Class 

On  the  Curve  ym—G[x)~Oy  and  its   Associated 

Abelian  Integrals. 


DISSERTATION 


PRESENTED  TO  THE   BOARD  OF  UNIVERSITY  STUDIES  OF  THE 

JOHNS  HOPKINS  UNIVERSITY  FOR  THE  DEGREE  OF 

DOCTOR  OF  PHILOSOPHY 


BY 


WILLIAM  H.  MALTBIE, 

Baltimore. 


1894. 


• '"of  rwt 


PRESS    OF    THE    FRIEDENWALD    COMPANY, 
BALTIMORE,    M  D. 


INDEX. 

PAGE 

Introduction 1 

I.  Bi-rational  Transformation  of  the  Curve  ym —  6r  (x)  =  0 1 

II.  The  Curve  ym  —  Bmn  (x)  =  0.    Its  Genus.    Its  Multiple  Points 

AND  THEIR  EQUIVALENT  NUMBER  OF  DOUBLE  POINTS  AND  CUSPS,      3 

III.  The  most  General  Rational  Function  of  x  and  y 8 

IV.  Integrals  Connected  with  the  Curve  f=0.    Their  Reduc- 

tion to  Two  Standard  Types 10 

V.  Integrals  of  the  First  Kind.    Their  Number 13 

VI.  Periods  of  Integrals  of  the  First  Kind.    Their  Form,  and 

Reductions  in  their  Number 18 

Biographical  Sketch 26 


INTRODUCTION. 

The  well-known  work  of  Clebsch  and  Gordon  on  the  Abelian  Functions 
assumes  that  the  fundamental  curve 

f(xy)  =  0 

is  such  that  only  two  of  the  values  of  y  permute  at  each  branch  point,  and  that 
the  multiple  points  are  either  cusps  or  multiple  points  with  distinct  tangents. 
There  exists,  however,  a  large  class  of  curves  included  in  the  form 

y™—G(a?)  =  0 

(where  G(x)  is  a  rational  function  of  x)  which  violate  both  these  hypotheses. 
They  may,  it  is  true,  be  made  to  satisfy  these  hypotheses  by  subjecting  them  to 
a  set  of  bi-rational  transformations,  but  in  the  process  they  are  deprived  of  all 
their  simplicity. 

The  purpose  of  this  paper  is  to  make  a  direct  investigation  of  the  integrals 
associated  with  these  curves  and  at  the  same  time  retain  their  characteristic 
simplicity  of  form. 

I  desire  here  to  express  my  gratitude  and  my  sincere  appreciation  of  the 
great  kindness  and  assistance  of  Professor  Craig  and  of  Professor  Franklin,  not 
only  in  the  preparation  of  this  paper,  but  throughout  my  entire  residence  at 
the  Johns  Hopkins  University. 


I. 

Bi-rational  Transformation  of  the  Curve  ym  —  G(x)=zO. 

Consider  the  algebraic  equation 

*,-—  G(aO  =  0, 
and  write  it  in  the  form 

*m-!t)=o>  (1) 

where  Bg  (x)  and  B  (x)  are  rational  entire  functions  of  x  of  degree  q  and  r 


respectively.     On  this  form  Osgood  *  has  made  the  following  reduction.     Put 

y'  «' 

and  (1)  becomes 

y,m  Br  (aft)  —  tm  +  r-<iBq  (a/Q  =  0 . 
Put  now 

t=zaxf  +  bt'  V=l] 

and  we  have 

y>™Br(x')  —  (ax/  +  b)  "+'-«22ff(a/)=:  0 .  (2) 

Again,  let 


RrW 

and  we  have 

y"m-\Br(x')\'-'Bm  +  r^)  =  0; 

or,  grouping  together  such  factors  of  the  product 

as  occur  to  the  mth  degree, 

y"m—  {Bl(x')\mBmir+1).mi  (*)  =  0. 
If  finally  we  put 

and  drop  the  accents,  we  have 

ym  —  Bm{v  +  1)(x)=0,  (3) 

where  v  is  an  integer  >0. 

This  is  Osgood's  reduction.     Examination  of  (2)  will  show,  however,  that 
when 

the  function  Bm  +  r  (a/)  is  no  longer  entire,  and  a  different  reduction  must  be 
made  for  this  form.     The  successive  transformations 


y»'   '  -Rq{x')' 
give  us  the  form 

or  as  before,  if 

ym-Bml,_1)(x)  =  0,  (4) 

*  Dissertation,  Gottingen,  1890. 


where  fi  is  an  integer  ]>  1.     All  possible  curves  of  the  form  (1)  are  therefore 
reduced  by  bi-rational  transformation  to  the  form 

ym  —  Bmn(x)  =  0.  »>0.  (5) 

The  important  things  to  note  are  :  1.  That  the  degree  of  the  polynomial 
R  is  a  multiple  of  the  exponent  of  y,  and  2.  that  not  more  than  m  —  1  of  the 
linear  factors  of  R  can  coincide. 


II. 

The  Cueve  ym  —  Rmn(x)  =  0.    Its  Genus.    Its  Multiple  Points  and 
their  Equivalent  Number  op  Double  Points  and  Cusps. 

We  write  the  curve  (5)  in  the  form 

ym  —  (x  —  a1)(x  —  a2)     .     .     .     {x  —  amn)  —  0,  (6) 

where  the  a's  are  real  or  imaginary  constants  not  more  than  m  —  1  of  which 
can  be  equal.  These  a's  are  all  branch  points  of  the  function,  and  around 
each  of  them  all  the  m  values  of  y  permute  in  one  or  more  cycles,  according 
as  the  number  of  coincident  a's  is  or  is  not  prime  to  m.  Moreover,  since  the 
a's  are  mn  in  number,  it  follows  that  the  point  infinity  is  not  a  branch  point  of 
the  function.* 

Introducing  homogeneous  co-ordinates,  we  have 

iz=mn 

f=zmn-mym—Y[(x  —  aiz)  =  0;  (7) 

and  from  this  form  we  see  at  once  that  no  term  is  of  degree,  in  x  and  z  jointly, 
less  than  m(n — 1);  i.  e.,  the  curve  has  at  x  =  z-=zO  an  m(n — l)-ple  point, 
whose  tangents,  as  may  easily  be  seen,  are  all  coincident. 

Again,  the  curve  (7)  has  no  other  multiple  points  except  such  as  arise  from 
the  coincidence  of  two  or  more  a's.  For  the  necessary  and  sufficient  conditions 
that  the  point  0/3/2/  shall  be  a  double  point,  and  therefore  the  necessary  con- 
ditions that  it  be  a  multiple  point  of  higher  order,  are 

J-./VyV)  =  0, 

|/(!B'y*')=o. 

*  Forsyth.    Theory  of  Functions,  p.  160. 


These  three  conditions  are  equivalent  to  the  two  following, 

2/m(n-l)-iym_.Qj 
i  =  mn 

^(x  —  a1z)(x  —  a2z)  ....  (x  —  ai_1z)(x-—ai^iz)  ....  (x  —  amnz)  ==  0 , 

i  =  l 

which  can  be  satisfied  only  by  x'zziz'  =  0  (the  multiple  point  at  infinity),  or 
when,  in  connection  with  an  identity  of  one  or  more  other  a's  with  a0  we  have 
x!  =  a^. 

If  then  the  a's  are  all  distinct,  the  curve  will  have  only  the  one  multiple 
point  x  =  z  z=  0 ,  and  its  genus*  will  be 

(mn — l)(mn — 2)       ™  ,QX 

2>=v ^ ;  —  #,  (8) 

where  E  denotes  the  number  of  double  points  and  cusps  to  which  the 
m(n  —  1  )-ple  point  is  equivalent.  The  coincidence  of  the  tangents  at  this 
multiple  point  makes  the  ordinary  relation,  "A  &-ple  point  is  equivalent  to 
J&  (k  —  1)  double  points,"  invalid ;  and  we  must  seek  other  means  for  the 
evaluation  of  E. 

Assume  all  the  a's  to  be  distinct.  The  function  has  then  mn  branch  points 
at  each  of  which  all  the  values  of  y  permute  in  a  single  cycle,  and  these  are  the 
only  branch  points.     The  genus  of  the  curve  is  therefore 

(m  —  1)  (mn)  —  2m  +  2 (m  —  1)  (mn  —  2)  ,Q>. 

P—                    2                     ~~                2               '  {  } 
and  we  have 

E=m(mn—  2)(n—  1)  ^  ^ 

When,  however,  some  of  the  a's  coincide,  the  resulting  multiple  points 
have  common  tangents,  the  formula  \h  (k  —  1)  again  fails,  and  the  method 
above  employed  gives  us  only  the  total  number  of  double  points  and  cusps  to 
which  the  two  multiple  points  are  equivalent.  We  must  accordingly  make  a 
direct  investigation  of  these  points,  and  determine  whether  the  number  of 
double  points  and  cusps  to  which  they  are  equivalent  is  variable  with  the  a's. 

Making  y  =  0  the  line  at  infinity,  and  returning  to  the  non-homogene- 
ous form,  the  equation  of  the  curve  becomes 


vm(n  —  1) 


—  It  (x  —  a(z)  =  0.  (11) 


*The  mathematical  faculty  of  the  Johns  Hopkins  University  have  agreed  to  use  the 
term  "genus"  in  place  of  "deficiency." 


The  multiple  point  is  now  at  the  origin,  and  the  line  zz=0  is  the  common 
tangent.  For  the  evaluation  of  the  multiple  point  we  will  follow  the  method 
introduced  by  Cayley,*  and  form  by  Newton'sf  method  the  expansion  for  each 
branch  in  the  neighborhood  of  the  origin, 

z=zAxa+Bx^  + , 

where  A,  B,  .  .  .  .  are  constants  and 

i<«</3<r<  — 

To  determine  A  and  a  we  substitute  in  (11) 

z  z=  Axa  , 
which  gives  us  the  form, 

Jm(n-l)x*m(n-l)  _xmn  _|_  A Z  (d^  Xmn ~l  +  a  —  A2 Z  (a^)  &>*-*  +  **+.   §   §   >  _>  Q  . 

The  term  xmn  has  its  exponent  independent  of  a ,  and  none  of  the  following 
exponents  can  be  made  equal  to  it  so  long  as  we  have  «^>1.  Therefore  we 
must  have 

am(n  —  1)  — mn, 
n 
n  —  1 
and 

Ip-rri 

Am*-l*=zlM   A^e^^K        p=zt,2,&, m(n  —  1) . 

The  curve  has  evidently  m(n  —  1)  branches  corresponding  tothem(n — 1) 
values  of  A.     For  the  farther  development  of  any  branch  we  assume 

2p7ri  n 

z  z=z  em(n—1>  xn— i  +  Bx$ , 
and  again  substitute  in  (11).     The  result  will  be 

Xmn  +  m(n—  1)  B&^^)  X^l[m{n~l)~l]  +  fi    +.   .   .   .   +  Bm{n-1]  X^{n-\)_xmn 

+  Z{ai)\J^-^)  xmn-l+^rx-\-Bxmn-l±P\ =  0. 

Now  the  term 

Z{ai)e^{n-\)  xmn-1  +  ^r1 

has  its  exponent  independent  of  ft  and  none  of  the  following  exponents  can  be 
made  equal  to  it  so  long  as  /9  ^>  — — —  ^>  1 ;  and  the  least  of  the  preceding  expo- 

*  Collected  Mathematical  Papers,  Vol.  V,  p.  520. 
t  As  given  by  Salmon,  Higher  Plane  Curves,  p.  44. 


nents  is  — ^r  [m(n  —  1)  —1]  +  /5.     We  have  therefore 

4pTTt 

n^+J.        -R__—I{ai)e^^-i) 
P  —  n—l'  m(n—  1) 

If,  as  can  easily  be  shown,  the  terms  of  the  series  all  have  n  —  1  as  the 
common  denominator  of  their  exponents,  then  each  of  the  branches  of  the  curve 
is  an  (n  —  l)-tic  branch;  i.  e.,  consists  of  n  —  1  partial  branches  given  by  the 
developments 

2(P)  _-.  e2nilm(n-l)  +  r^IJ  x^=l  _         ^  (ai)       ^  [m  (n - 1)  +  ,T=1 J  ^T\  -}-...., 

1  m(n 1) 

.r        p           ■       2     "I        n  y  ,      \  f       2p  .       4    "|       n-fl 

g(p)  e       L"1  (n— 1)  **"  n— lJ^n  — 1 ^  W       e       Lm(»  — l)  +  n  — lJ   ^.n  —  1     I 

2  m(n  —  1)  *  '  *  *  ' 


n_1  ra(w — 1) 

The  ensemble  of  partial  branches  belonging  to  the  m(n  —  1)  total  branches 
will  be  obtained  by  giving  to  p  in  the  above  system  the  values  1,  2,  3  ...  . 
m(n  —  1) .     If  now  we  form  all  possible  differences 

Z8  Z8'      t 

and  denote  by  M  the  sum  of  the  exponents  of  their  first  terms,  we  may,  following 
Cayley,  say  that  our  m  (n — l)-ple  point  is  equivalent  to  }[if — 3m  (n — l)(w — 2)] 
double  points  and  m  (n  —  l)(w  —  2)  cusps. 

The  evaluation  of  M,  while  possible  in  any  particular  example,  is  entirely 
too  complicated  to  be  attempted  in  the  general  case.  The  important  thing  to 
notice  is  that  since  the  expansions  for  the  partial  branches  differ  only  in  the 
exponents  of  e ,  the  exponents  of  the  first  terms  of  the  differences  z{8r)  —  2#"} 
will  be  independent  of  the  relative  values  of  the  a's ;  and  therefore  that  the 
number  of  the  double  points  and  cusps  to  which  the  m(?i — l)-ple  point  is 
equivalent  is  unchanged  when  two  or  more  of  the  a's  coincide.  If  then  this 
number  can  be  found  when  all  the  a's  are  distinct,  it  is  found  for  all  cases. 
But  this   has   already   been   done.     We   can    therefore   now  affirm  that  the 

m(n — l)-ple  point  is  equivalent  to  — ± ^  '  ordinary  double  points 

and  cusps. 


If  h  (k  <  m)  of  the  a's  coincide,  the  method  of  page  4  gives  us  the  com- 
bined equivalence  of  the  two  multiple  points;  and  the  permanence  of  the 
equivalence  of  the  first  enables  us  to  find  at  once  the  equivalence  of  the  second. 

In  particular,  if  Bmn  (x)  has  a  factor  of  the  form  (x  —  a*)*,  where  k  is 
prime  to  m ,  we  will  denote  its  equivalence  by  JS^ .     Then 


P 


(mn  —  l)(mn  —  2) m(n  —  l)(mn  —  2) ™ 


But  the  function  has  now  mn  —  k  +  1  branch  points,  around  each  of  which  all 
the  values  of  y  permute  in  a  single  cycle.     Its  genus  is  therefore 


P 


__(m  —  l)(mn  —  &  +  !)  —  2  m  +  2 
__  _ 


.    „       (m -!)(&- 1) 

MV  2  '  (12) 

On  the  other  hand,  if  k  is  not  prime  to  m,  let  k=.lp ,  m=zXp  ,  where  I  is 
prime  to  X,  Then  the  function  will  have  mn —  k  branch  points  at  which  all 
the  values  of  y  permute  in  a  single  cycle,  and  one  where  they  permute  in  p 
cycles.     Its  genus  will  therefore  be 


(m  —  l)(mn  —  k)  +  m  —  p  —  2m  +  2 
and  we  have 


*=  2 


E,  =  (fn-m-l)  +  P-l  >  (13) 

Moreover,  a  second  Newton  expansion  will  show  that  the  equivalence  of  a{  is 
unaffected  by  the  relative  value  of  the  remaining  a's.  We  are  therefore  able  to 
find  the  equivalence  of  all  the  multiple  points  of 

ym  _  (J,.  __  a^kx  ^  __  a2)*3 fa  __  a^fca  =  0  }  (14) 

where 


S?  k{  ss  mn 


i=l 


III. 


The  Most  General  Rational  Function  of  x  and  y. 


The  most  general  rational  function  of  x  and  y,  when  they  are  connected  by 
the  relation  (5),  is  of  the  form 


A'tf*-1  +  B'ym-2  + +I/y  +  M' 

Ay"1'1  +  Bym~2  + +  Ly  +  M    ' 


(15) 


where  A',  E,  ....  M',  A,  B,  .  .  .  .  if,  are  arbitrary  rational  entire  func- 
tions of  x.  The  first  reduction  to  be  made  is  to  render  the  denominator  a 
function  of  x  alone. 

When  m  =  2  we  have  the  hyperelliptic  case.     When  m=3,  Thomae* 
makes  the  reduction  by  multiplying  both  numerator  and  denominator  by 

(Atfr*  +  Byv  +  0)(Ayh  +  Byz*  +  C) , 


where 


2ni 

„  3 


In  dealing  with  the  general  case  we  may  either  extend  this  method  of 
Thomae's,  or  follow  the  method  used  in  the  general  theory  of  Abelian  Func- 
tions.    For  this  denote  ym  —  Rmn{%)  by /and 

Ay™-1  +  Bym~2  + +  Ly  +  M  by  <p . 

Then  the  product  of  <p  by  a  factor  which  renders  it  rational  in  x  alone  is,  as  is 
well  known, 


1      0      0     0     0 0     —  R       0     0     0.. 

0      10     0     0 0  0    —  R    0     0.  . 


fym~3 


ABODE M        0        00    0...;^w_1 

m  — 2 


0    A    B     C    D L        M       000 


n 


00     00     0 A        B      QBE 


*  Ueber  eine  specielle  Klasse  Abelscher  Functionen.    Halle,  1877. 


9 


Multiply  the  first  column  by  ym  and  add  it  to  the  (m  -|~  l)th  column.  Multiply 
the  second  by  ym  and  add  it  to  the  (ra  +  2)nd,  etc.  Then,  since /=0  we 
have  for  our  rationalizing  factor 


J  = 


M 

Aym 

Bym  .  .  . 

.     Jy™ 

Kym 

ym-1 

L 

M 

Aym  .  . 

.  ijr 

Jym 

ym  —  2 

K 

L 

M      .  . 

.  Hym 

lym 

ym  —  3 

0    D 

E 

M 

Aym 

y' 

B     G 

D 

L 

M 

y 

A    B 

0 

K 

L 

l 

a  determinant  of  order  m  whose  law  of  formation  is  evident. 

The  same  result  may  be  obtained  by  assuming  a  factor  of  the  form 

performing  the  multiplication,  and  equating  to  zero  the  coefficients  of 

Vi  y\y* ym"1 

which  will  give  us  m  —  1  homogeneous  equations  for  the  determination  of  the  m 
quantities  «,  /9,  y,  .  .  .  .  fi . 

All  of  the  methods  indicated  must  give  (except  for  a  possible  factor  in  x 
alone)  the  same  result.  For  given  any  two  factors  F(xy)  and  $(xy)  which 
will  reduce  <p  (xy)  to  a  function  of  x  alone,  we  have  at  once 

<p(xy)  F{xy)z=(/;1{x)) 
<p{xy)  0(xy)  =  <ff2(x), 


from  which 


w-gg#<w. 


The  most  general  rational  function  of  x  and  y,  when  they  are  connected 
by  the  relation  /  (xy)  =.  0 ,  will  be  of  the  form 


where  Al}  A2,  .  .  .  .  Am>  and  the  product  d(p  are  functions  of  #  alone. 


10 


IV. 


Integrals  Connected  with  the  Curve /=z0.    Their  Reduction  to 

Two  Standard  Types. 

All  possible  integrals  connected  with  the  curve  /=.  0  may  now  by  the  use 

of  the  multiplier  J  %-  be  reduced  to  the  form 
dy 

{Al3r^+A^r^+....+AmdX9  (16) 

J  rf 

dy 
where  At  and  X  are  rational  entire  functions  of  x. 

From  this  point  we  have  two  analogies  to  follow.  The  curve 
ym  —  Bmn(x)=z0  evidently  occupies  a  middle  ground  between  the  hyperelliptic 
curve  y2 —  Ri(x)=zQ,  on  the  one  hand,  and  the  general  Abelian  curve 
F(xy)=z0  on  the  other. 

We  will  follow  first  the  analogy  of  the  hyperelliptic  integrals  and  reduce 
the  general  integral  (16)  to  integrals  of  two  more  simple  types. 

The  theory  of  decomposition  of  rational  fractions  enables  us  at  once  to 
reduce  (16)  to  a  sum  of  integrals  of  the  two  forms 

fP* (%>**,  and  f-J2lM*!L 

dy  dy 

where  Px  and  cpx  are  rational  entire  functions.  These  in  turn  by  simple  separa- 
tion of  their  terms  and  use  of  the  relations  f~0,  and  §£-  =  my™-1 ,  are 
reduced  to  the  two  forms 

[Q{x)yadx  m)  f     <p{x)y«dx  nft. 

J    my™-1     '         {     }  iix—aymy™-1'         {i*} 

where  a  is  a  positive  integer  less  than  m. 

Consider  the  form  (17).  Let  the  degree  of  Q(x)  be  denoted  by  q,  and  let 
L  (x)  be  a  rational  entire  function  of  degree  I.     If  then  we  subtract  from 

(Q(x)y*dx 
J    my™-1 

the  expression  d  (L  (x)  ya  +  1), 

we  shall  reduce  the  integral  to  an  integrated  part  and  to  the  new  integral 


11 

The  relation  /=  0  gives  us 

dy  _,  R'{x) 
dx        mym~1> 

and  by  means  of  this  and/=zO  we  can  put  the  integral  in  the  form 

[lQ(x)-mLi{x)R{x)-(a+\)L{x)R>(x)-\y«  ^ 
J  my™—1 

If  now  we  take  I  =  q  — mn  +  1  >  the  last  two  terms  in  the  bracket  become  a 
polynomial  in  x  of  degree  q.  We  can  now  so  determine  the  q  —  mn  -f-  2 
arbitrary  constants  in  this  polynomial  that  the  entire  expression  in  the 
brackets  becomes  a  polynomial  in  x  of  degree  q  —  q-\-mn  —  2  =  mw  —  2 .  If 
now  we  separate  the  terms  of  this  polynomial,  we  shall  reduce  all  the  integrals 
of  the  form  (17)  to  the  form 

[x^yadx         /3<mn — 2  ,-.qx 

J  mym~x  '       a<mn —  1  *     ' 

and  these  we  will  call  integrals  of  the  first  type. 

The  form  (18)  divides  into  two  classes  according  as  a  is  or  is  not  a  root  of 
Rmn  (x)  =  0  .  When  Rmn  a^O,  we  subtract  from  the  integral  (18)  an  expres- 
sion of  the  form 

where  G  is  an  undetermined  constant.  The  form  (18)  then  reduces  to  an  inte- 
grated part  and  to  the  integral 

rr        ,    ''■  C(a  +  l)ya(x  —  a)^fLdx  —  (J—  l)Ch*+ldx 

[f    <p(x)yadx  \   ~r   jj  \         J  dx  \        J    y 

J  L{x  —  a)lmym-1  {x  —  a)1 

__  tt<p{x)—  C(a  +  1)  R'(x)  (x  —  a)  +  (I—  1)  CmR  (a;)]  y«  dx 
J  (x  —  a)lmym~l 

If  now  C  be  determined  by  the  relation 

<p  (p)  +  (I  —  1)  CmR  (a)  =  0 , 

the  above  integral  reduces  to  one  of  the  form 

[        0(x)yadx 
}(x  —  a)l-1mym-i; 

and  this  process  may  evidently  be  continued  till  l=z  1 ,  i.  e.  all  the  integrals 
of  the  form  (18)  reduce,  when  Rmn(a)  ^p  0,  to  the  form, 


<F{ifi-i-  «^™-i-  (2°) 

(x  —  a)  mym    x  ^ 


These  we  shall  call  integrals  of  the  second  type. 


12 

If,  however,  a  is  a  root  of  B  (x)  =  0  of  order  h ,  we  subtract  from  the  inte- 
gral (18)  an  expression  of  the  form 


d(       ^a+1       ) 


,(x  —  a) 
C(a  +  IX*—  a)  y  \Mdx--  0(1  +  k  —  1)  y+1  <& 

^  («  —  a)l+kmym-1 

Put  now 

i2(a0  =  (s  —  aYGx{x), 
R>(x)  =  (x  —  af-^G^x), 

and  this  fraction  becomes 

[g(g  +  1)  ^  (a?)  -  <?(*  +  ft  -  1)  gg,  fefl  ^ 

We  are  therefore  able  to  reduce  the  form  (18)  to  an  integrated  part  and  to  the 
integral 

La  [yW-g^  +  ilgiW  +  g  +  fe-il^W)]  dx 

)y  '  (x—aymy™-1 

Since  G1  (a)  zj=  0 ,  and  Gr2  (a)  ;£  0 ,  we  can  so  determine  G  by  the  relation 

p  (a)  —  0{  («+  1)  G2{a)  +  (l  +  k  —  l)m01  (a)}  =  0 
that  this  integral  shall  reduce  to  the  form 

f       F(x)  ya  dx        t 
](x  —  a)l-lmym-i; 

and  the  process  may  evidently  be  continued  till  we  are  reduced  to  the  form 

[F{x)yadx 
J    my™-1 

In  this  we  may  reduce  the  degree  of  F  by  the  method  already  given  and  find 
again  only  integrals  of  the  first  type. 

All  the  integrals  connected  with  the  curve  f=zO  are  now  reduced  to  the 
two  types 

I      [rfjfdx  /3<mw-  2       ] 


IL     f     F(x)y*dx  R(a)  zfz  0 

}(x  —  a)mym-1'     a<m  —  l 


13 


y. 

Integeals  of  the  Fiest  Kind.    Theie  Numbee. 
Consider  the  type  I  and  put  it  in  the  form 

m  —  1  ' 

)m(R(x))  m 
When  x  becomes  very  great  this  becomes  comparable  to 

L^  +  an-n(m-l)^? 

which  remains  finite  for  x  very  great  so  long  as  /3  +  an — n(m —  1)<C — 1« 
This  inequality  for  ft  positive  is  possible  for  all  values  of  a  <  m  —  1.  In  order 
therefore  to  find  the  number  of  integrals  of  the  first  type  which  remain  finite 
for  x  very  great,  we  have  the  formula 

a  =  m  — 2 

Vr                           n      (m—l)(mn — 2) 
/ \mn — an — n  —  1J  =  - -^ -  • 

a  =  0  * 

The  hyperelliptic  analogy  would  lead  us  to  call  these  lm        )\ — "—J  integrals, 

"integrals  of  the  first  kind,"  but  the  analogy  fails  at  this  point.  In  the 
hyperelliptic  case  m  =  2,  Rt (x)  is  therefore  reducible  by  bi-rational  transfor- 
mation to  the  form 

f  —  R2n(x)  =  0> 

where  R2n(x)hsiS  no  multiple  factors.  The  curve  has  therefore  no  multiple 
points  except  the  2(n  —  l)-ple  point  at  x=zz=zO.  At  this  the  integrals 
remain  finite,  and  therefore  the  hyperelliptic  integral  which  remains  finite  for  x 
very  great  remains  finite  throughout  the  entire  plane. 

In  our  case,  if  x  —  a  is  a  multiple  factor  of  Rmn(x) ,   the  expression 

— \^_x  is,  in  general,  infinite  of  an  order  \1  at  the  point  xz=a;  and  the  corre- 
sponding integral  is  therefore  not  of  the  first  kind.  If,  however,  Rmn  (x)  has 
no  repeated  factors,  we  have  * —        '  ^ ~^—L  integrals  of  the  first  kind,  and 

this  is  equal  to  the  genus  of  the  corresponding  curve /=0  . 

If  a  is  a  root  of  Rmn(x)=zO  of  order  k,  we  take  the  general  integral  of 
the  first  type 

(Q(x)yadx .  (21) 

J    my"1'1    '  v     ; 


14 

and  subtract  from  it 

At 

,{x — a) 


d(  (¥+1    \ 


C(a  +  l)y(x— a)d-ldx  —  0(k—  l)y«  +  'dx 

(XX 

(x  —  af 

_  y  [(7(g  +  1)  (x  —  a)  R'—  Cm  (k  —  1)  #]  da; 
~"  (a?— aYmtf*-1  ' 

which,  if  we  define  (xt  and  6r2  as  before,  is  equal  to 

y"  \_C(a  +  1)  &2  (a?)  —  Cm  {h  —  1)  Gt  (x)  ]  cfc 

If  now  C  be  defined  by  the  relation 

Q  (a) -<?[(«  +  1)  G, (a)  +  m  (h- 1)  fi^a)]  =  0 , 

our  integral  reduces  to  an  integrated  part  and  to  an  integral  of  the  form 

Mx  —  a)  Qi(x)yadx 
J  mym-1  ' 

where   Qx  (x)  is  of  the  degree  mn  —  3 .      This  reduction   may  evidently  be 
continued  till  we  have  the  form 

[(x—af-1Qk_l(x)yadx 
J  my™-1 

Denote  now  by  the  symbol  [2]  ,  where  z  is  a  real  number,  integer  or  fractional, 
the  greatest  integer  contained  in  z.     We  note,  first,  that 


*_i5[A(m_i_a)], 


and  therefore,  second,  that  all  our  integrals  of  the  first  type  reduce  to  a  set  of 
integrals  of  the  form 

J  mym-l-a  * 

which  are  finite  when  x-=za .     Separate  this  into  its  terms,  put  m  —  1  —  a=id , 
and  we  have  the  integrals 

Ux  —  a)^xfidx  ^mn~ 2~ [£J 

J  mf  0<^ro  —  1 


15  X^ 

In  order  that  these  may  remain  finite  for  x  very  great  we  must  have 

The  number  of  integers  satisfying  this  condition  is 


=1 


When  k  is  prime  to  m  a  known  theorem  gives  us 

8  =  m  —  1 

(h  —  l)(m—l)  (22) 


m 


2 


a  quantity  which  we  have  already  found  as  the  equivalence  of  a  &-ple  point  on 
f=  0 ,  when  k  is  prime  to  m. 

If  k  is  not  prime  to  m,  put  h  =  fy?  and  m  =  X[>  and  an  immediate  extension 
of  the  above-mentioned  theorem  gives  us 


\  =  m  —  1 

yp<P__(&  — l)(m  —  l)  +  /o  —  1 

5  =  1"- 


2-  -  ,  (23) 


which  is  the  equivalence  of  the  &-ple  point  when  k  is  not  prime  to  m. 

If  similar  reductions  are  made  for  all  the  multiple  factors  of  R  (x) ,  the 
number  of  integrals  which  remain  finite  throughout  the  plane,  i.  e.  the  number 
of  integrals  of  the  first  kind,  will  evidently  be 


(^■2)(»-l),^i=J>|  (24) 


2 

where  %Et  is  the  sum  of  the  double  points  and  cusps  equivalent  to  those 
multiple  points  of  f=.0  which  result  from  repeated  factors  of  R(x),  The 
number  of  integrals  of  the  first  kind  is  therefore  in  all  cases  equal  to  the 
genus  of  the  curve. 

We  have  now  formed  p  integrals  of  the  first  kind,  and  they  are  evidently 
linearly  independent.  But  if  we  ask  for  the  most  general  form  of  an  integral 
of  the  first  kind,  and  whether  there  may  or  may  not  be  more  than  p  of  them, 
we  must  turn  our  attention  to  the  more  general  theory  of  Abelian  Functions 


16 

and  follow  the  analogy  which  it  presents.     We  know  that  the  most  general 
integral  connected  with  the  curve  fz=.  0  is  of  the  form 

f    JMT-'-Mar  >+■    -+A,  _.*,.  (25) 


(x  —  a^i  (x  —  a2)?a  .  .  .  .  (x  —  av)lvmyr 


What  are  the  conditions  under  which  this  will  always  remain  finite  ? 

Consider  the  point  x=nai.  The  integral  is  evidently  infinite  at  this  point 
unless  the  numerator  vanish  also.  Two  cases  arise  corresponding  to  the  two 
conditions  R  («J  zfz  0 ,  and  R  (at)  ss  0 . 

When  R {at) zfzO ,  y  has  for  xzzL^m  values  all  different  and  all  different 
from  zero.  In  order  then  that  the  integral  remain  finite  when  x-=.ai  it  is 
necessary  and  sufficient  that  the  numerator,  a  polynomial  in  y  of  degree  m  —  1 , 
vanish  for  m  different  values  of  y.  Its  coefficients  must  therefore  all  vanish, 
i.  e.  the  A's  must  all  have  a  factor  x —  at)  and  the  integral  reduces  to  the  form 


Atf*- 1  +  ....  +  4 


I (x  —  a^  ....(#  —  0-i)li~l ....  (a?  —  a„)lvmy- 


m  —  1 


dx . 


A  repetition  of  the  argument  will  evidently  remove  from  the  denominator  all 
the  factors  (x —  at) ,  where  R  (a{)  :£  0  . 

When  at  is  a  root  of  R  (x)  =  0  of  order  k ,  we  write 

R(x)  =  (x—ai)kG(x)) 
and  put  for  y  its  value 

(R  (x))m  =z(x  —  at)m  ( G  {x))m. 

The  integral  now  becomes 

k(m  —  1)                tn  —  1                                     fc  (m  —  2)                 m  — 2 
f^l(g  —  «,)       »        (G(x))m      +A2(X—0.1)      rn        (g  fr))    m      + 

J        (x  —  aj* (x—  «,)i+  ^  -  ;....(a?  —  a,)V(0(aO)  " 

In  order  that  this  remain  finite  for  x=zaiit  must  reduce  to  the  form 

(<p  (x)  dx 

where  <p  (x)  is  finite  for  x-=zaiJ  and  k  is  less  than  unity.     The  numerator  must 

therefore  contain  (x  —  atY    L     m    J  as  a  factor.     For  this  it  is  necessary  and 
sufficient  that  we  have 


17 


A1=i(x—ai)liBl9 
A2=(x  —  at)       lmJB2, 

Am  =  (x-ai)      L    -     lBm. 


(27) 


The  repetition  of  this  process  will  remove  from  the  denominator  of  (25)  all  the 
factors  (x —  «f),  where  R(ai)=:0;  and  (25)  accordingly  takes  the  form 

Ullr-*  +  A4r-  +  ....  +  Am<lx.  (28) 

J  my™'1  v 

This  must  still  be  examined  for  x  very  great  and  for  those  values  of  x 
which  satisfy  mym~1  =  0 .  For  x  very  great  we  see  at  once  that,  if  (28)  is  to 
remain  finite,  we  must  have 

At=0,  ~\ 

A%  of  degree  n  —  2 
J  (29) 

Am  of  degree  (m  —  1)  n  —  2 .  J 

The  points  which  satisfy  mym~1=z  0  are  the  roots  of  R  (x)  z=  0 .  If  a«  be 
a  simple  root  of  R  (x)  =  0 ,  the  integral  reduces  to  the  form 

r       £  (x)  dx  R  (x)  =z(x  —  at)  G  (a?) 

and  is  therefore  finite.  If  all  the  roots  of  R(x)  are  simple,  the  most  general 
integral  of  the  first  kind  will  be  of  the  form  (28) ,  the  degrees  of  the  A'b  will  be 
given  by  (29),  and  the  number  of  arbitrary  constants  included  in  the  integral  is 
therefore 


2((i-D»-i)=('Mt-aj(w-1) 


t=2 


If  at  is  a  Mold  root  of  R  (x)  =  0 ,  we  have  only  to  put  lt  =  0  in  the  dis- 
cussion above  in  order  to  show  that  the  number  of  constants  is  reduced  by 

^      —    ,  which  we  know  to  be  equal  to  the  number  of  double  points  and 

cusps  equivalent  to  the  multiple  point  at  on  the  curve /=  0  .     The  number  of 
arbitrary  constants  in  the  most  general  integral  of  the  first  kind  is  therefore 

(mn-2)(m  —  l)_IJE_ 
2  *     F 


18 

In  the  more  general  theory,  based  on  the  curve  <p  (xy)  =  0  of  degree  m  in  x 
and  y,  the  general  integral  of  the  first  kind  is  found  to  be  of  the  form 

(Q(xy)dx 

i      <P'v        ' 

where  Q  is  of  degree  m  —  3  in  both  x  and  y ,  and  has  the  &-ple  points  of  <p  =  0 

as  (k — l)-ple  points.     Q  has  therefore  * ^ —        '   coefficients  which  are 

subject  to  Iat  ^  ~~~^  conditions  (a^^the  number  of  i-ple  points  of  <pz=zO). 

Consequently,  if  there  is  no  reduction  in  the  number  of  these  conditions,  there 
will  be  p  and  only  p  integrals  of  the  first  kind  linearly  independent.  But  in 
order  to  show  that  there  are  only  p  we  must  show  that  no  such  reduction  takes 
place.  In  the  case  we  have  been  considering  we  find  not  only  the  number  of 
the  conditions  but  the  conditions  themselves.  We  thus  at  once  see  that  there 
is  no  reduction  in  their  number,  and  affirm  that  there  are  p  and  only  p 
linearly  independent  integrals  of  the  first  kind. 


VI. 

Periods  of  Integrals  of  the  First  Kind.    Their  Form,  and 
Reductions  in  their  Number. 

If  we  ask  after  the  number  and  character  of  the  periods  of  these  integrals 
of  the  first  kind,  we  can  proceed  in  two  ways.  We  can,  in  the  first  place,  form 
the  Rieman  surface  connected  with  the  curve /=0.  It  will  be  m  sheeted  and 
of  genus  p.  We  know  that  by  bending  and  stretching  it  can  be  deformed  into 
a  sphere*  with  p  handles,!  and  therefore  that  the  integral  of  a  uniform  function 
has  on  it  2p  periods. 

In  general  these  2p  periods  are  distinct,  i.  e.  there  exists  between  them 
no  linear  homogeneous  relation  with  integer  coefficients.  The  proof  of  this 
last  statement  need  not  be  given  here  ;  as  it  is  the  same  as  in  the  case  of  the 
general  Abelian  Functions ;  except  that,  in  the  consideration  of  the  inequality 


\  XdY >0,t 


*  Jordan,  Liouville's  Journal,  Series  2,  t.  XI  (1866). 

f  Klein.  On  Rieman's  Theory  of  Functions.  Section  8  of  Miss  Hardcastle's  transla- 
tion. 

JPicard,  Traits  d' Analyse,  Tome  2,  pp.  405-409.  X-\-iY  is  an  integral  of  the  first 
kind.  iT  is  the  contour  made  up  of  the  p  cuts  through  the  holes,  the  p  cuts  around  the 
holes,  and  the  p  —  1  cuts  joining  these  into  a  continuous  contour. 


19 

we  note  that,  in  the  region  of  the  branch  points,  -=-  is  no  longer  of  the  order 
- — ,  but  of  the  order  ^-r  in  the  case  of  a  simple  branch  point :  and  of  the  order 

»7~i  (*  <C m)  m  *ne  case  °f  a  multiple  branch  point,     (r  is  the  radius  of  the 

circle  of  integration  about  the  branch  point.) 

We  might  proceed  to  form  a  system  of  normal  integrals  and  to  discuss  the 
relations  existing  among  their  periods :  but  the  discussion  is  so  strictly  analogous 
to  that  usually  given  for  the  general  Abelian  Functions  that  we  prefer  to  turn 
to  the  system  of  integrals  already  formed,  and  consider  the  form,  and  certain 
reductions  in  the  number,  of  their  periods. 

We  assume,  first,  that  all  the  factors  of  R  (x)  ==  0  are  distinct.  Take  upon 
the  plane  an  arbitrary  point  u0 ,  and  draw  from  it  loops  to  all  the  mn  branch 
points  of  /  (i.  e.  the  roots  of  R  (x)  =:  0) .  Denote  the  loop  around  at  by  a] 
when  it  is  described  in  the  positive  direction,  and  by  of*  when  it  is  described 
in  the  negative  direction. 


■2niS 


Take  now  any  one  of  our  integrals  of  the  first  kind 

^r  =  [<p(x)dx  ^ 
J    my8 

The  effect  on  Us  of  a  small  circle  around  at  is  to  multiply  it  by  e   m    ==>* 
The  value  of  U  along  the  straight  line  from  u0  to  the  small  circle  about  at 
will  be  denoted  by  At . 

The  inverse  function  y  will  have  as  periods  the  values  of  Us  along  any 
contours  which  return  y  to  its  original  value.  Among  such  contours  we 
choose  the  following, 

«1         «2  >  al        «3  9  al        «4  ,  .  .  .  .  «i         amn  , 


20 

and  denote  the  corresponding  periods  by 

co2 ,  ws ,  .  . . ,  wmn . 
Consider  the  first  of  these.     We  have  evidently 

(o2=:A1{l—^m-1^)+A1(^-1^  —  ^m-^)  + +A1(X*—l)+A2(X—l). 

If  now  we  multiply  this  by  Xs ,  we  have  a  new  period  Xsw2  to  which  corresponds 
the  contour  af-2^^.  In  the  same  way  we  have  X28co2  corresponding  to 
af  _8«2«i,  etc.  Treating  the  other  co's  in  the  same  way,  we  have  the  following 
table  of  periods  associated  with  Us , 


co2,     Xsw2y     X™<o2, X(m-V&a)2, 

co5 ,     Xsco3 ,     X28w3 , ^m  "  V  sco3 , 

«i4,     Xscoi}     X28co4, X{m-1)8wif 


<»mn  ,   XS0)mn  ,    X28COmn  ,     .  .  .  X{t 


■1)8, 


(30) 


We  say  farther  that  this  table  of  periods  is  complete,  i.  e.  that  all  possible 
periods  associated  with  Us  can  be  expressed  as  linear  homogeneous  functions, 
with  integer  coefficients,  of  the  periods  (30). 

In  order  to  show  this  we  note : 

1°  The  most  general  period  of  y  corresponds  to  a  contour  made  up  of 
km  -\- 1  loops  described  in  the  same  direction,  and  /  loops  described  in  the 
opposite  direction,  where  h  and  I  may  have  any  positive  integer  values  including 
zero.     (The  same  loop  described  r  times  is  regarded  as  r  loops.) 

2°  Between  any  two  loops  of  any  period  we  may  introduce  the  nugatory 
contour  afcq~k,  corresponding  to  the  period  0 . 

3°  Given  a  general  period  wx  and  its  corresponding  contour,  there  exists 
a  period  Xucox ,  whose  contour  consists  of  the  same  loops  arranged  in  the  same 
cyclic  order  as  in  the  contour  belonging  to  cox ,  but  beginning  at  an  arbitrary 
point  in  the  cycle.  (This  is  merely  a  generalization  of  what  has  already  been 
done  in  the  case  of  w2,  Xsw2,  ....  X^m-1^w2.) 

4°  The  new  periods,  —  a)t  +  cok ,  izfzlc,  corresponding  to  the  new 
contours  aflak;  together  with  the  period  w2>  which  corresponds  to  both  the 
contours  «f_1«;>  and  aj^a^y  enable  us  to  replace  the  last  loop  of  any  contour  by 
any  other  loop. 

1°,  2°,  3°,  and  4°  being  granted,  let  it  be  required  to  form  from  the 
periods  (30)  an  arbitrary  period  corresponding  to  a  contour,  denoted  by  2), 
consisting   of  km  + 1  positive  and   /   negative    loops.     (The   argument   will 


21 

be  the  same  for  hm-\-l  negative  and  I  positive  loops.)  To  do  this  repeat  the 
contour  corresponding  to  co2  k  times ;  and  from  this,  by  the  introduction  of  the 
proper  nugatory  contours  «f«7"*>  we  ge*  a  new  contour  A  which  has  the  same 
number  of  positive  and  negative  loops  as  D  and  in  the  same  order.  By 
successive  cyclic  permutation  of  the  loops  of  A ;  and  the  addition,  after  each 
permutation,  of  the  proper  one  of  the  periods  deduced  in  4°;  A  becomes 
identical  with  D ;  and  the  corresponding  period  is  given  as  a  linear  homogeneous 
function,  with  integer  coefficients,  of  the  periods  (30).  The  system  (30)  is 
therefore  complete. 

There  are,  however,  certain  reductions  among  the  periods  (30).  We 
know  that  the  value  of  Us  along  a  contour  composed  of  all  the  loops  is  zero. 
But  this  contour,  by  a  process  entirely  analogous  to  that  used  in  the  case  of  D 
and  A  y  may  be  reduced  to  the  contour  corresponding  to  io2  repeated  n — 1 
times  and  followed  by  wi .  Moreover,  the  periods  used  in  this  reduction  are 
none  of  them  derived  from  o)t .  We  may  therefore  express  cot  in  terms  of  the 
other  periods,  and  strike  out  from  the  table  30  the  row  of  periods 


a)iyX8o)ifX2So)iy A<« 


1)8, 


The  remaining  mn —  2  periods  in  the  first  column  of  (30)  are  in  general 
distinct,  since  each  of  them  contains  an  A  that  does  not  appear  in  any  of  the 
others. 

If  d  is  prime  to  m ,  the  table  (30) ,  by  virtue  of  the  relation  xm  =  1 ,  and 
by  a  permutation  of  the  columns,  takes  the  form 


>2,  Xco2,  Fw2,       tmJo>2, 

>s ,  *co3  >  ^3  f         .  .  .  .  Am     ^3  , 


(0mn-l>  ^wmn-\y  *  ^mn-1  y  •  •  •  «*m       ^r 


(31) 


(We  have  chosen  the  last  row  as  the  one  to  be  dropped.)     If  m  is  odd,  we  have 
between  the  periods  of  any  row  the  one  relation 


k  =  m  —  1 
Oh 


>£p  =  0 


fc=0 


We  may  therefore  strike  out  any  column  (say  the  last).  The  remaining  periods 
are  in  general  distinct;  and  the  integral  has,  under  the  hypotheses  made 
above  as  to  d  and  m,  the  maximum  number  of   independent  periods,  i.  e. 

(mn  —  2)  (m —  1)  =  2p . 


22 


If  m  is  even  (d  still  prime  to  m)  we  have  the  relation 


/12-  =  _1 


A2     =  —  A«. 


?/i 


When  -g  is  odd,  we  can  express  all  of  the  even  powers  of  X  in  terms  of  the  odd 
powers;  and  the  table  of  periods  (31)  may  be  replaced  by  the  table 


Xto2, 
Xcoz, 


Xzco2y 
XhoB , 


.•.'.* 


CO 


i) 


~) 


^°mn-li  A  C0mn_ly 


(32) 


m, 


But  X  is  now  a  primitive  — th  root  of  unity,  and  therefore 


2 


x+xB+x5+ +>-i=yA»+i==o1 


k  =  0 


We  may  therefore  drop  the  last  column  of  (32),  and  the  integral  has  now  only 
(mn  —  2)  I — —  1 )  periods,  which  are  in  general  distinct.     (We  note  that  this  is 

m 

twice  the  genus  of  the  curve  y*  —  R  /m-v2n(a;)  ■=.  0 ,  all  the  factors  of  B  being 
distinct.) 

If,  on  the  other  hand,  -    is  even,  we  can  drop  half  the  columns  of  (31) ;  but 

we  know  of  no  relation  connecting  those  which  remain.     The  integral  has  in  this 

case  (mn  —  2)  =-  periods  which  are  in  general  distinct. 

If  d  is  not  prime  to  m ,  put  S=zs/j.  and  m  =  ?y* ,  where  s  is  prime  to  r . 
We  have  then  the  relation  XrS  =  1 ,  and  from  this  the  relation 

k=m—'l  k  =  r  —  l 

2>lM  =  ;/£>>. 

fc=0  fc=0 

The  table  of  periods  (30)  accordingly  takes  the  form 

co2,         Xsco2,         X28co2,         ^"^Vo        1 


':;> 


XS(D* 


\02 


«>mn-l,  ^V 


mn  —  1  f 


>a> 


mn  — 1  > 


.  ..AC-1* 


.  .  Ji'-1)** 


'mn  —  1  •  J 


(33) 


«F^ 


2niS 


But  /I5  =  e"»»~  =  e"  =  /58 ,  where  /?  is  a  primitive  rth  root  of  unity.  Since  r  is 
prime  to  s  and  /9r  =  1 ,  (33)  may,  by  use  of  this  value  of  ^  and  proper  permu- 
tation of  the  columns,  be  put  in  the  form 


co2 


CO 


3> 


(0mn  —  1 
k=r  —  1 


>{>< 


'mn  —  1) 


F 


fr-xm. 


mn  —  1 


(34) 


When  r  is  odd,  2J*k  ~  0 ,  we  strike  out  the  last  column,  and  have  in  general 


fc  =  0 


(mn —  2)(r  —  1)  independent  periods.     (We  note  that  this  is  twice  the  genus 
of  the  curve  y2  —  Brt,n  (x)  =  0 ,  all  the  factors  of  B  (x)  being  distinct.) 

If  r  is  even,  a  discussion  exactly  analogous  to  that  made  in  the  case  when 

m  was  even  and  d  prime  to  m  shows  that  we  have  in  this  case  (mn  —  2)  f  -^  —  l) 

or  (mn —  2)     independent  periods,  according  as  ^  is  odd  or  even. 
We  may  tabulate  all  these  results  as  follows : 

m  odd,  the  no.  of  periods  is  (mn  —  2)(m — 1). 


d  prime  to  m 
and 


d  =  sfi.  m  =  r/u. 

s  prime  to  r 

and 


odd, 


even, 


r  odd,     " 
"oodd,     « 


2  even, 


(mri_2)g_l). 

(mn-2)g). 

(mn—  2)(r—  1). 
(TO»_2)g-l). 

(mn-2)(0. 


The  meaning  of  a  portion  of  these  reductions   is  very  evident.     When 
d=zs/jt  and  m=.r/jty  the  integral 


x^dx 


m(Rmn(x)} 


m         y 


x^dx 


s   ' 

m  (Bmn  (x))'r 


which  last  is  an  integral  connected  with  the  curve  yr  —  Bmn  (x)=zO.     I  am  able 
at  present  to  offer  no  satisfactory  explanations  of  the  other  reductions.    We 


24 

have  said  that  the  periods,  after  the  above  reductions  have  been  made,  are 
in  general  distinct.  It  is  evident  that  the  only  farther  reductions  which 
can  arise,  so  long  as  the  factors  of  R(x)  are  distinct,  must  come  from  relations 
among  the  a»'s  themselves.  It  may  be  possible  to  so  choose  the  roots  of 
R  (x)  =  0  that  some  at  least  of  the  integrals  connected  with  the  curve  shall 
have  less  than  ran  —  2  distinct  co's.  For  example,  the  integral  of  the  first 
kind, 

[xn~Hx 

connected  with  the  curve 

ym  —  (xn—a^)(xn  —  a^) (xn  —  a%)  =  0 

reduces  to  the  integral 


[dx 


connected  with  the  curve 

yrn_^_an)(x_Qn)  _  (*_<)  =  0, 

when  we  take  x11  as  our  new  variable ;  and  the  new  integral  has  only  ra  —  2  w's. 
A  similar  case  is  the  reduction  of  the  integrals  connected  with  the  sextic 

y^  —  (xi  —  al)(x2  —  ai)(x2~(4)  =  0 

to  elliptic  integrals.* 

It  is  evident,  however,  that  no  farther  reductions  can  take  place  among 
the  periods  derived  from  anyone  co ,  so  long  as  the  factors  of  R(x)  are  distinct. 
Therefore,  while  in  the  case  of  the  hyperelliptic  integrals  we  can  only  say  that 
there  are  at  least  two  periods,  we  are  able  in  the  present  case  to  say  that  the 
integral 

(xPdx  d=zs/i 


has  at  least  (ran  —  2)  ( ^  —  1 )  distinct  periods. 


We  have  limited  ourselves  so  far  to  the  case  where  all  the  factors  of  R  (x) 
are  distinct.  The  case  where  some  of  them  are  the  same  presents  no  insuper- 
able difficulties,  but  introduces  a  great  deal  of  complexity.  We  shall  limit 
ourselves  to  a  simple  case. 

Suppose  h  (k<^m)  of  the  factors  of  R (x)  to  be  of  the  form  (x  —  a}),  the 
others  remaining  distinct.     We  shall  have  in  this  case  mn  —  2  —  h  co's  corre- 

*Picard,  Traits'  d' Analyse,  Tome  I,  p.  217. 


25 

sponding  to  contours  of  the  form  «f_1«o  and  one,  coj ,  corresponding  to  a 
contour  of  the  form  aj*_1«f  .  If  k  is  prime  to  ra ,  the  point  at  will  accordingly 
give  rise  to  m  —  1  periods  and  we  have  in  all  (ran — k —  l)(ra —  l)=z2p 
periods. 

If  k  is  not  prime  to  ra ,  put  k  =  lp  and  m-=.Xp ,  and  by  the  argument 
used  when  d  was  not  prime  to  ra  we  see  that  the  point  ctj  gives  rise  to  X  —  1 
periods,  and  we  have  in  all  (ran  —  k  —  2)  (ra  —  1)  +  ^  —  1 « 

The  number  of  periods  is  in  this  case  therefore  (/>  —  1)(X  —  1)  less  than  the 
maximum  2p .  The  number  of  periods  in  both  these  cases  will  of  course  be 
subject  to  reduction  when  ra  is  even  or  when  d  is  not  prime  to  ra . 

We  have  been  speaking  so  far  of  curves  reduced  to  the  standard  form 

ym  —  Bmn(x)=zO, 

but  similar  relations  exist  among  the  periods  associated  with  the  curve, 

ym—Rs{x)  =  0, 

where  R  is  rational  and  entire  and  s  is  any  integer.  The  difference  in  the 
discussion  will  arise  from  the  fact  that  the  point  infinity  is  now  a  branch  point 
where  all  the  values  of  y  permute  in  one  or  more  cycles.  We  will  have  then  a 
similar  complete  system  of  periods  cot ,  and  their  multiples  by  rath  roots  of 
unity. 

In  particular,  if  we  make  ra  zz:  3  and  s  =  4 ,  we  have  for  the  three  integrals 
connected  with  the  curve  yzz=.x(x  —  a)(x  —  b)(x  —  t) ,  the  periods  given  by 
Picard,* 


ho[,  Xco{f,  ho[", 


Comptes  Rendus,  Tome  93,  p.  835. 


Biographical  Sketch. 

The  author,  William  H.  Maltbie,  was  born  in  Toledo,  Ohio,  August  26, 
1867.  He  was  under  the  care  of  private  instructors  and  in  various  elemen- 
tary and  high  schools  until  1885,  at  which  time  he  entered  the  Ohio  Wesleyan 
University  at  Delaware,  Ohio,  from  which  institution  he  received  the  degree  of 
A.  B.  in  1890,  and  of  A.  M.  in  1892.  In  1890  he  was  appointed  Professor  of 
Mathematics  in  Hedding  College  at  Abingdon,  111.  In  1891  he  entered  the 
Johns  Hopkins  University  as  a  candidate  for  the  degree  of  Doctor  of  Philo- 
sophy, selecting  Mathematics  as  his  principal  subject,  with  Astronomy  as  first 
and  Physics  as  second  subordinate.  In  January,  1893,  he  was  appointed 
University  Scholar,  and  in  June,  1894,  Fellow  in  Mathematics. 


un  rv 

OF  V      J 


/ 


